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Derivatives of sin, cos, e and ln

Calculus ¡ 21 problems, easy to harder

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How we do it

  1. The inside
  2. The outside
  3. The inside's slope
  4. Multiply
  5. Check

The steps for Derivative of sin(3x). Every solution names its steps, so you always know what comes next.

Worked examples

Derivative of sin x Easy

The question

Question: find the derivative of f(x) = sin x.

f(x) = sin x

1 The rules

The derivative of sin x is cos x.

sin x → cos x

2 Each term

Use the rule on each term.

f′(x) = cos x

3 Tidy up

So f′(x) = cos x.

= cos x

4 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives 0.5403, with a calculator in radians.

f′(1) = cos 1 ≈ 0.5403

Now the slope between two very close points: x = 1 and x = 1.001.

A calculator gives about 0.54.

Very close to 0.5403.

Right!

f(1.001) − f(1)0.001 ≈ 0.54

Let's look back.

First: the rules.

Then: each term.

Then: tidy up.

Last: check.

Answer
= cos x

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Derivative of sin(3x) Medium

The question

Question: find dy/dx for y = sin(3x).

y = sin(3x)

1 The inside

There is something inside: 3x.

Call it u.

u = 3x

Then y is just sin u.

y = sin u

2 The outside

The derivative of sin x is cos x.

sin x → cos x

First the outside.

The rule on sin u gives cos u.

dydu = cos u

3 The inside's slope

Now the inside.

The power rule on 3x gives 3.

dudx = 3

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

dydx = dydu × dudx

Put the inside back in place of u.

= cos(3x) × 3

Careful!

Do not stop here!

We must multiply by the inside's slope, 3.

Careful: not cos(3x) ✗

Write the numbers in front.

So dy/dx = 3 cos(3x).

= 3 cos(3x)

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives −2.97, with a calculator in radians.

dy/dx at x = 1: −2.97

Now the slope between two very close points: x = 1 and x = 1.001.

A calculator gives about −2.97.

Very close to −2.97.

Right!

y(1.001) − y(1)0.001 ≈ −2.97

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer
= 3 cos(3x)

▶ Watch it solved

Derivative of ln(x^2 + 1) Harder

The question

Question: find dy/dx for y = ln(x² + 1).

y = ln(x2 + 1)

1 The inside

There is something inside: x² + 1.

Call it u.

u = x2 + 1

Then y is just ln u.

y = ln u

2 The outside

The derivative of ln x is 1 over x.

ln x → 1x

First the outside.

The rule on ln u gives 1 over u.

dydu = 1u

3 The inside's slope

Now the inside.

The power rule on x² + 1 gives 2x.

dudx = 2x

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

dydx = dydu × dudx

Put the inside back in place of u.

= 1x2 + 1 × 2x

Careful!

Do not stop here!

We must multiply by the inside's slope, 2x.

Careful: not 1x2 + 1 ✗

So dy/dx = 2x/(x² + 1).

= 2xx2 + 1

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives 1, with a calculator.

dy/dx at x = 1: 1

Now the slope between two very close points: x = 1 and x = 1.001.

A calculator gives about 1.

Very close to 1.

Right!

y(1.001) − y(1)0.001 ≈ 1

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer
= 2xx2 + 1

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Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. derivative of sin x
    Answer
    = cos x
  2. derivative of 3cos x
    Answer
    = −3 sin x
  3. derivative of e^x + x^2
    Answer
    = ex + 2x
  4. derivative of 5ln x
    Answer
    = 5x
  5. derivative of 2sin x - 4cos x
    Answer
    = 2 cos x + 4 sin x
  6. derivative of 4e^x - 3ln x
    Answer
    = 4ex − 3x
  7. derivative of x^3 + 2sin x
    Answer
    = 3x2 + 2 cos x
  8. y = 6cos x - e^x, find dy/dx
    Answer
    = −6 sin x − ex

Medium

  1. derivative of sin(3x)
    Answer
    = 3 cos(3x)
  2. derivative of cos(2x)
    Answer
    = −2 sin(2x)
  3. derivative of e^(2x)
    Answer
    = 2e^(2x)
  4. derivative of e^(-x)
    Answer
    = −1e^(−x)
  5. derivative of 4sin(5x)
    Answer
    = 20 cos(5x)
  6. derivative of ln(3x)
    Answer
    = 1x
  7. derivative of cos(x^2)
    Answer
    = −2x sin(x2)
  8. derivative of e^(x^2)
    Answer
    = 2x e^(x²)

Harder

  1. derivative of ln(x^2 + 1)
    Answer
    = 2xx2 + 1
  2. derivative of sin(x^2 + 1)
    Answer
    = 2x cos(x2 + 1)
  3. derivative of cos(3x + 1)
    Answer
    = −3 sin(3x + 1)
  4. derivative of 3e^(x^2 + 1)
    Answer
    = 6x e^(x² + 1)
  5. derivative of ln(2x + 5)
    Answer
    = 22x + 5